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Logic
Introduction to Logic
What is logic?
• Logic is the study of valid reasoning.
• That is, logic tries to establish criteria to
decide whether some piece of reasoning is
valid or invalid.
• OK, so then what do we mean by ‘valid
reasoning’?
Reasoning
• A piece of reasoning consists of a sequence
of statements, some of which are claimed to
follow from previous ones. That is, some
are claimed to be inferred from others.
• Example: “Either the housemaid or the
butler killed Mr. X. However, if the
housemaid would have done it, the alarm
would have gone off, and the alarm did not
go off. Therefore, the butler did it.”
Valid Reasoning
• While in every piece of reasoning certain
statements are claimed to follow from others, this
may in fact not be the case.
• Example: “If I win the lottery, then I’m happy.
However, I did not win the lottery. Therefore, I am
not happy.”
• A piece of reasoning is valid if the statements that
are claimed to follow from previous ones do
indeed follow from those. Otherwise, the
reasoning is said to be invalid.
Sound Reasoning
• Not all valid reasoning is good reasoning.
• Example: “If I win the lottery, then I’ll be
poor. So, since I did win the lottery, I am
poor.”
• This piece of reasoning is valid, but not
very good, since it assumed an absurd claim
(‘If I win the lottery, I’ll be poor.’ Huh??)
• Sound reasoning is valid reasoning based on
acceptable assumptions.
Truth and Implication
• Logic studies the validity of reasoning.
• Logic does not study soundness.
• Therefore, logic alone cannot tell us
whether an argument is good. Hence, logic
alone is not a guide to truth.
• Instead, logic can tell us, assuming certain
things to be true, what else will be true as
well. Thus, logic is a guide to implication.
Arguments, Premises and
Conclusion
• In logic, pieces of reasoning are analyzed using
the notion of an argument
• An argument consists of any number of premises,
and one conclusion
• Again, in logic, we are merely interested in
whether the conclusion follows from the premises:
we are not interested in whether those premises
are true or acceptable.
Deductive Validity vs
Inductive Validity
• An argument is said to be deductively valid if,
assuming the premises to be true, the conclusion
must be true as well.
• An argument is said to be inductively valid if,
assuming the premises to be true, the conclusion is
likely to be true as well.
Argument Forms
• “If I win the lottery, then I am poor. I win the
lottery. Hence, I am poor.”
• This argument has the following abstract structure
or form: “If P then Q. P. Hence, Q”
• Any argument of the above form is valid, including
“If flubbers are gook, then trugs are brig. Flubbers
are gook. Hence, trugs are brig.”!
• Hence, we can look at the abstract form of an
argument, and tell whether it is valid without even
knowing what the argument is about!!
Formal Logic
• Formal logic studies the validity of
arguments by looking at the abstract form of
arguments.
• Formal logic always works in 2 steps:
– Step 1: Use certain symbols to express the
abstract form of premises and conclusion.
– Step 2: Use a certain procedure to figure out
whether the conclusion follows from the
premises based on their symbolized form alone.
Example Step 1: Symbolization
• Use symbols to represent simple propositions:
– H: The housemaid did it
– B: The butler did it
– A: The alarm went off
• Use further symbols to represent complex claims:
– H  B: The housemaid or the butler did it
– HA: If the housemaid did it, the alarm would go off
– ~A: The alarm did not go off
Example Step 2: Symbolization
• Transform symbolic representations using
basic rules that reflect valid inferences:
1. H  B
A.
2. HA
A.
3. ~A
A.
4. ~H
2, 3 MT
5. B
1, 4 DS
Propositional Logic
• Propositional Logic studies validity at the level of
simple and compound propositions.
• Simple proposition: An expression that has a truth
value (a claim or a statement). E.g. “John is tall”
• Compound proposition: An expression that
combines simple propositions using truthfunctional connectives like ‘and’, ‘or’, ‘not’, and
‘if … then’. E.g. “John is tall and Mary is smart
Predicate Logic
• Predicate Logic extends Propositional Logic
by adding individuals, predicates, and
quantifiers
• Individuals: ‘John’, ‘Mary’
• Predicates: ‘tall’, ‘smart’
• Quantifiers: ‘all’, ‘some’
Uses of Formal Logic
• Evaluation/Checking:
– Formal logic can be used to evaluate the validity of
arguments.
• Clarification/Specification:
– Formal logic can be used to express things in a precise
and unambiguous way.
• Demonstration/Proof:
– Formal logic can be used to figure out what follows
from a set of assumptions.
• Computation/Automated Reasoning:
– Formal logic can be used for machine reasoning.
The Method of Formal Logic
• Step 1: Use certain symbols to express the
abstract form of premises and conclusion.
• Step 2: Use a certain procedure to figure out
whether the conclusion follows from the
premises based on their abstract
symbolizations.
FOL
• FOL (the language of First-Order Logic):
The formal language that we use to
symbolize statements
Individual Constants
• An individual constant is a name for an
existing object.
• Examples: john, marie, a, b
• Each name is assumed to refer to a unique
individual, i.e. we will not have two objects
with the same name.
• However, each individual object may have
more than one name.
Predicates
• Predicates are used to express properties of
objects or relations between objects.
• Examples: Tall, Cube, LeftOf, =
• Arity: the number of arguments of a
predicate (E.g. Tall: 1, LeftOf: 2)
Atomic Sentences
• Combining one predicate with the proper
number of individual constants yields an
atomic sentence.
• Examples: Tall(john), LeftOf(a,b), a=a
• Prefix notation: the predicate precedes the
individual constant(s). E.g. Tall(marie)
• Infix notation: the predicate is in between
the individual constants. E.g. a=b
Argument, Premises, Conclusion,
Validity and Soundness
• An argument consists of 0 or more premises and 1
conclusion.
• An argument is (deductively) valid iff:
– assuming its premises are true, its conclusion must be
true as well.
• An argument is (deductively) sound iff:
– it is valid, and
– its premises are true.
• In pure logic, we are only interested in validity.
Demonstrating Invalidity
• An argument is valid if it is impossible for the
conclusion to be false while the premises are true.
• Thus, to demonstrate invalidity, all we have to do
is to demonstrate that it is possible for the
conclusion to be false while the premises are true.
• The easiest way to do this is to come up with a
scenario (or possible world) in which all premises
are true and the conclusion false.
Demonstrating Validity
• To demonstrate validity, we have to show that there is no
possible way for all premises to be true and the conclusion
false all at the same time.
• Showing a scenario in which all premises are true, and in
which the conclusion is true as well, does not demonstrate
validity, b/c there may still be a different scenario in which
all premises are true and the conclusion false.
• Obviously, this holds true in general. Hence, showing one
possible scenario cannot demonstrate validity!
• Of course, we could try and generate all relevant possible
worlds, but this method is either impractical (there are too
many), or simply impossible (there are infinitely many).
Step-by-step Reasoning
• OK, so what do we do? Well, we can do what we
do in everyday reasoning: we start with the
premises, and we gradually work our way to the
conclusion: “Either the housemaid or the butler
killed Mr. X. Now, we know that if the housemaid
would have done it, the alarm would have gone
off. But, the alarm did not go off. Therefore, the
housemaid did not do it. So, since it was either the
housemaid or the butler, it must have been the
butler.”
Structure of the Argument
• 1. Either the housemaid or the butler did it.
• 2. If the housemaid did it, the alarm would
have gone off.
• 3. The alarm did not go off.
• 4. (2+3) The housemaid did not do it.
• 5. (1+4) The butler did it.
Intermediate Results
• In the previous proof, claim 4 (the housemaid did
not do it) is called an intermediate result (or subconclusion).
• Intermediate results reflect steps in our reasoning.
• Intermediate results are important, b/c:
– If each of the steps is valid, then the reasoning as a
whole is valid as well (i.e. conclusion validly follows
from the premises).
– The step-by-step reasoning counts as a proof if each of
the steps is obviously valid.
Proofs
• A proof is a sequence of statements, starting
with premises, followed by intermediate
conclusions, and ended by the conclusion,
where each of the intermediate conclusions,
and the conclusion itself, is an obvious
consequence from (some of) the premises
and previously established intermediate
conclusions.
Formal Proofs
• Formal proofs try to formalize proofs by:
– Symbolizing the statements in a proof (again,
we will use FOL for this)
– Spelling out what we count as an ‘obvious
consequence’ based on this symbolization
What is ‘obvious’ is not obvious
• Problem: ‘obvious’ is a bit of a vague term, as
what is obvious to some, may not be obvious to
others. So, what are going to count as ‘obvious’?
• We are going to play it safe: In formal proofs, we
are only going to allow steps that are about as
obvious as we can get. Thus, we are only going to
allow ‘baby inferences’.
• In formal proofs, bigger inferences, which may
still be obvious to many (if not all of us), will still
have to be broken up into smaller ones!
Inference Rules
• Inference rules formalize these ‘baby inferences’.
• Example: An inference rule may indicate that if
you have a statement of the form ‘a=b’ then you
can infer a statement of the form ‘b=a’. Notice:
– This inference rule is purely symbolic/syntactic/formal
– This inference rule reflects an obvious inference
• Inference rules may need any number of
statements from which the new statement is
inferred (though with too many statements, the
rule may no longer be considered ‘obvious’).
– Most inference rules require one or two statements.
– Some inference rules require no statements at all.
Formal Systems
• There are many formal systems of logic, each with
their own predefined set of inference rules:
– First of all, the nature of the inference rules depends on
the symbols that the system uses to express statements.
– Moreover, even if two systems use the same symbols,
they may still have different inference rules.
Connectives: From Atomic
Claims to Complex Claims
• So far, we have only seen atomic claims:
claims consisting of a single predicate. E.g.
“a is to the right of b”.
• Boolean connectives: and, or, not, if …
then, if and only if
• We can combine atomic claims using
boolean connectives to form complex
claims. E.g. “Either a is to the right of b or a
is to the left of b”.
Propositional Logic
• Propositional Logic is the logic involving
complex claims as constructed from atomic
claims and boolean connectives.
Truth-Functional Connectives
• Boolean Connectives are usually called
truth-functional connectives.
• The truth value of a complex claim that has
been constructed using a truth-functional
connective is a function of the truth value of
the claims that are being connected by that
connective.
Negation
• The claim “a is not to the right of b” is a complex
claim. It consists of the atomic claim “a is to the
right of b” and the truth-functional connective
“not”.
• We will call the above statement a negation.
• To express negations, we use the symbol ‘’
• ‘’ should be put in front of what you want to be
negated.
• Thus, the above statement will be symbolized as:
RightOf(a,b)
Truth-Table for Negation
• ‘’ is truth-functional, since the truth-value
of a negation is the exact opposite of the
truth-value of the statement it negates.
• We can express this using a truth table:
P P
T F
F T
Conjunction
• The claim “a is to the right of b, and a is in
front of b” is called a conjunction.
• The two claims that are being conjuncted in
a conjunction are called its conjuncts.
• To express conjunctions, we will use the
symbol ‘’
• ‘’ should be put between the two claims.
• Thus, the above statement will be
symbolized as: RightOf(a,b)  FrontOf(a,b)
Truth-Table for Conjunction
• ‘’ is truth-functional, since a conjunction is
true when both conjuncts are true, and it is
false otherwise.
• Again, we can show this using a truth table:
P
T
T
F
F
Q PQ
T
T
F
F
T
F
F
F
Disjunction
• The claim “a is to the right of b, or a is in
front of b” is called a disjunction.
• The two claims that are being disjuncted in
a disjunction are called its disjuncts.
• To express disjunctions, we will use the
symbol ‘’
• ‘’ should be put between the two claims.
• Thus, the above statement will be
symbolized as: RightOf(a,b)  FrontOf(a,b)
Truth-Table for Disjunction
• ‘’ is truth-functional, since a disjunction is
true when at least one of its disjuncts is true,
and it is false otherwise.
• Again, we can show this using a truth table:
P
T
T
F
F
Q PQ
T
T
F
T
T
T
F
F
Combining Complex Claims:
Parentheses
• Using the truth-functional connectives, we
can combine complex claims to make even
more complex claims.
• We are going to use parentheses to indicate
the exact order in which claims are being
combined.
• Example: (P  Q)  (R  S) is a conjunction
of two disjunctions.
Parentheses and Ambiguity
• An ambiguous statements is a statement whose
meaning is not clear due to its syntax. Example :
”P or Q and R”
• In formal systems, an expression like P  Q  R is
simply not allowed and considered unsyntactical.
• Claims in our formal language are therefore never
ambiguous.
• One important application of the use of formal
languages is exactly this: to avoid ambiguities!
Exclusive Disjunction vs
Inclusive Disjunction
• Notice that the disjunction as defined by ‘’
is considered to be true if both disjuncts are
true. This is called an inclusive disjunction.
• However, when I say “a natural number is
either even or odd”, I mean to make a claim
that would be considered false if a number
turned out to be both even and odd. Thus, I
am trying to express an exclusive
disjunction.
How to express Exclusive
Disjunctions
• We could define a separate symbol for exclusive
disjunctions, but we are not going to do that.
• Fortunately, exclusive disjunctions can be
expressed using the symbols we already have:
(PQ)  (PQ)
P
T
T
F
F
Q (P  Q)  (PQ)
T
T F F T
F
T T T F
T
T T T F
F
F F T F
!
Logically Equivalent Statements
• Two statements are logically equivalent if
their truth-conditions are identical.
• Simply put, two statements are logically
equivalent if it is impossible for one
statement to be true while the other is false.
• To express that two statements P and Q are
logically equivalent, we will write: PQ
Some Important Equivalences
• Double Negation:
–
PP
• DeMorgan:
–
–
(P  Q)  P  Q
(P  Q)  P  Q
• Distribution:
–
–
P  (Q  R)  (P  Q)  (P  R)
P  (Q  R)  (P  Q)  (P  R)
Some Other Equivalences
• Commutation:
– PQQP
– PQQP
• Association:
– P  (Q  R)  (P  Q)  R
– P  (Q  R)  (P  Q)  R
• Idempotence:
– PPP
– PPP
Tautologies
• A tautology is a statement that is necessarily
true.
• Example: P  P
• Any statement that evaluates to True in
every row of its truth-table is a tautology.
Contradictions
• A contradiction is a statement that is
necessarily false.
• Example: P  P
• Any statement that evaluates to False in
every row of its truth-table is a
contradiction.
Equivalences
• Two statements are equivalent if they have
the exact same truth-conditions.
• Example: P and P
• In every row of their combined truth-table,
two equivalent statements are either both
true or both false.
Implication
• One statement implies a second statement if
it is impossible for the second statement to
be false whenever the first statement is true.
• Example: P implies P  Q
• In the combined truth-table, there is not a
single row where the implying statement is
true and the implies statement is false
Consistency
• A set of statements is consistent if it is
possible for all of them to be true at the
same time.
• Example: {P, P  Q, Q}
• In the combined truth-table of a consistent
set of statements there is at least one row
where they all evaluate to True.
Validity
• An argument is valid if it is impossible for
the conclusion to be false whenever all of
its premises are true.
• Example: P, P  Q  Q
• In the combined truth-table of a valid
argument, there is not a single row where all
premises are true and the conclusion is
false.
The Principle of Substitution of
Logical Equivalents
• Let us write S(P) for a sentence which has P as a
component part, and let us write S(Q) for the
result of substituting Q for P in S(P).
• The principle of substitution of logical equivalents
states that if P  Q, then S(P)  S(Q).
• Example:
– Since Small(a)  Small(a), it is also true that
(Cube(a)  Small(a))  (Cube(a)  Small(a))
Simplifying Statements
• Using the principle of substitution of logical
equivalents, and using the logical
equivalences that we saw before (Double
Negation, Association, Commutation,
Idempotence, DeMorgan, and Distibution),
we can often simplify statements.
• Example: (A  B)  A  (Commutation)
(B  A)  A  (Association)
B  (A  A)  (Idempotence)
BA
Negation Normal Form
• Literals: Atomic Sentences or negations thereof.
• Negation Normal Form: An expression built up
with ‘’, ‘’, and literals.
• Using repeated DeMorgan and Double Negation,
we can transform any truth-functional expression
built up with ‘’, ‘’, and ‘’ into an expression
that is in Negation Normal Form.
• Example: ((A  B)  C)  (DeMorgan)
(A  B)  C  (Double Neg, DeM)
(A  B)  C
Disjunctive Normal Form
• Disjunctive Normal Form: A disjunction of
conjunctions of literals.
• Using repeated distribution of  over , any
statement in Negation Normal Form can be
written in Disjunctive Normal Form.
• Example:
(AB)  (CD)  (Distribution)
[(AB)C]  [(AB)D]  (Distribution (2x))
(AC)  (BC)  (AD)  (BD)
Conjunctive Normal Form
• Conjunctive Normal Form: A conjunction of
disjunctions of literals.
• Using repeated distribution of  over , any
statement in Negation Normal Form can be
written in Conjunctive Normal Form.
• Example:
(AB)  (CD)  (Distribution)
[(AB)  C]  [(AB)  D]  (Distribution (2x))
(AC)  (BC)  (AD)  (BD)
Truth-Functional Connectives
• So far, we have seen one unary truth-functional
connective (‘’), and two binary truth-functional
connectives (‘’, ‘’).
• Later, we will see two more binary connectives
(‘’, ‘’)
• However, there are many more truth-functional
connectives possible:
– First of all, a connective can take any number of
arguments: 3 (ternary), 4, 5, etc.
– Second, there are unary and binary connectives other
than the ones listed above.
Truth-Functional Expressive
Completeness
• Since I can express any truth function using ‘’,
‘’, and ‘’, we say that the set of operators {,
, } is (truth-functionally) expressively complete.
• DeMorgan Laws:
– (P  Q)  P  Q
– (P  Q)  P  Q
• Hence, by the principle of substitution of logical
equivalents, since {, , } is expressively
complete, the sets {, } and {, } are
expressively complete as well!
Proofs
• Proof: A sequence of statements,
starting with zero or more assumptions,
where each of the statements is either
an assumption, or an obvious logical
consequence from (some of) the
assumptions and previously inferred
statements.
Proof by Contradiction
• ‘Assuming P to be the case, then I get some
kind of impossibility or contradiction.
Hence, contrary to my assumption, P cannot
be the case.’
• This pattern of reasoning is called Proof by
Contradiction (or Indirect Proof or Reductio
ad Absurdum or simply Reductio).
‘’
• The symbol ‘’ is used to express a logical
contradiction.
• Theorem: A statement  is a logical
contradiction iff   
• Theorem: A set of statements is logically
inconsistent iff {1, …, n}  
Soundness of Proof by
Contradiction
• Theorem:
– {1, …, n}   iff {1, …, n, }  .
• Proof: {1, …, n}   iff it is impossible for
each 1 to be true and  to be false iff it is
impossible for each 1 to be true and  to be
true iff {1, …, n, }  .
• So, once we show that, given initial assumptions
1, …, n , the further assumption  leads to a
contradiction (i.e. {1, …, n, }  ), we
know that  is a logical consequence from the
initial assumptions (i.e. {1, …, n}   ).
The Practice and Art of Formal
Proofs
• Given that formal proofs are not
mechanical, and thus require some amount
of creativity, the only way to learn to do
formal proofs is through practice, practice,
and more practice!
• Giving formal proofs is a skill and an art. In
fact, you will probably develop your own
style in making proofs.
The Material Conditional
• Let us define the binary truth-functional
connective ‘’ according to the truth-table below.
• The expression PQ is called a conditional. In
here, P is the antecedent, and Q the consequent.
P
T
T
F
F
Q PQ
T
T
F
F
T
T
F
T
‘If … then …’ Statements
• The conditional is used to capture ‘if … then …’
statements.
• Although the match isn’t perfect, most uses of ‘if
… then …’ are captured fine with the conditional.
• In particular, any ‘if … then …’ statement will be
false if the ‘if’ part is true, but the ‘then’ part false,
and the conditional captures this important truthfunctional aspect of any ‘if … then …’ statement.
‘If and only if’ and the
Material Biconditional
• A statement of the form ‘P if and only if Q’ (or ‘P
iff Q’) is short for ‘if P then Q, and if Q then P’.
Hence, we could translate this as (PQ) 
(QP). However, since this is a common
expression, we define a new connective ‘’:
P
T
T
F
F
Q PQ
T
T
F
F
T
F
F
T
Some Important Equivalences
• Implication:
– P  Q  P  Q
– (P  Q)  P  Q
• Transposition:
– P  Q  Q  P
• Exportation:
– P  (Q  R)  (P  Q)  R
• Equivalence:
– P  Q  (P  Q)  (Q  P)
– P  Q  (P  Q)  (P  Q)
Quantifiers
Introduction to Logic
From Propositional Logic to
Predicate Logic
• So far we have dealt with propositional logic
• The next step is to deal with predicate (or
quantificational) logic.
• Predicate logic builds on propositional logic.
Quantification: ‘All’ and ‘Some’
• In predicate logic, there are two quantifiers:
‘all’ and ‘some’.
• Here are some examples:
– x Mortal(x) ‘All things are mortal’
– x Mortal(x) ‘Some things are mortal’
– x (Human(x)  Mortal(x)) ‘Every human is
mortal’
– x (Human(x)  Mortal(x)) ‘Some human is
not mortal’
Parts of a Quantificational
Statement
• A quantificational statement such as
x(Human(x)  Mortal(x)) has the
following parts:
– Quantifier: In this case ‘’
– Variable: In this case ‘x’
– Well-formed formula, or wff: In this case
‘Human(x)  Mortal(x)’
From Propositional Statements to
Quantificational Statements
• Every quantificational statement can be
made from a propositional statement:
– Start with a propositional statement with
individual constants: Human(a)  Mortal(a)
– Change one or more individual constants into a
variable: Human(x)  Mortal(x)
– Quantify the atomic wff (which is not a claim)
to obtain a quantificational statement (which is
a claim): x (Human(x)  Mortal(x))
Free and Bound Variables and the
Scope of Quantifiers
• In the expression x (Human(x)  Mortal(x)),
the variable x that occurs in the atomic wff
Human(x)  Mortal(x) is bound by the quantifier
, as indicated by the ‘x’ right after the quantifier.
• The parentheses indicate the scope of the
quantifier.
• In the expression x (Human(x))  Mortal(x),
only the first ‘x’ is within the scope of the
quantifier, and thus bound. The second ‘x’ is free.
• Any expression with one or more free variables is
not a claim!
Universe of Discourse
• When we quantify, we usually have some
universe of discourse in mind. E.g. when I
say “Everyone did well on the homework”,
I am limiting myself to all students in this
class.
• When this is understood, I can simply write:
x (PerformedWell(x)).
• If not, I can always limit myself as follows:
x (Student(x)  PerformedWell(x)).